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A138530 Triangle read by rows: T(n,k) = sum of digits of n in base k representation, 1<=k<=n.

%I A138530 #20 Feb 16 2025 08:33:08
%S A138530 1,2,1,3,2,1,4,1,2,1,5,2,3,2,1,6,2,2,3,2,1,7,3,3,4,3,2,1,8,1,4,2,4,3,
%T A138530 2,1,9,2,1,3,5,4,3,2,1,10,2,2,4,2,5,4,3,2,1,11,3,3,5,3,6,5,4,3,2,1,12,
%U A138530 2,2,3,4,2,6,5,4,3,2,1,13,3,3,4,5,3,7,6,5,4,3,2,1,14,3,4,5,6,4,2,7,6,5,4,3,2,1
%N A138530 Triangle read by rows: T(n,k) = sum of digits of n in base k representation, 1<=k<=n.
%C A138530 A131383(n) = sum of n-th row;
%C A138530 A000027(n) = T(n,1);
%C A138530 A000120(n) = T(n,2) for n>1;
%C A138530 A053735(n) = T(n,3) for n>2;
%C A138530 A053737(n) = T(n,4) for n>3;
%C A138530 A053824(n) = T(n,5) for n>4;
%C A138530 A053827(n) = T(n,6) for n>5;
%C A138530 A053828(n) = T(n,7) for n>6;
%C A138530 A053829(n) = T(n,8) for n>7;
%C A138530 A053830(n) = T(n,9) for n>8;
%C A138530 A007953(n) = T(n,10) for n>9;
%C A138530 A053831(n) = T(n,11) for n>10;
%C A138530 A053832(n) = T(n,12) for n>11;
%C A138530 A053833(n) = T(n,13) for n>12;
%C A138530 A053834(n) = T(n,14) for n>13;
%C A138530 A053835(n) = T(n,15) for n>14;
%C A138530 A053836(n) = T(n,16) for n>15;
%C A138530 A007395(n) = T(n,n-1) for n>1;
%C A138530 A000012(n) = T(n,n).
%H A138530 Alois P. Heinz, <a href="/A138530/b138530.txt">Rows n = 1..141, flattened</a>
%H A138530 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DigitSum.html">Digit Sum</a>
%e A138530 Start of the triangle for n in base k representation:
%e A138530 ......................1
%e A138530 ....................11....10
%e A138530 ......... ........111....11...10
%e A138530 ................1111...100...11..10
%e A138530 ..............11111...101...12..11..10
%e A138530 ............111111...110...20..12..11..10
%e A138530 ..........1111111...111...21..13..12..11..10
%e A138530 ........11111111..1000...22..20..13..12..11..10
%e A138530 ......111111111..1001..100..21..14..13..12..11..10
%e A138530 ....1111111111..1010..101..22..20..14..13..12..11..10
%e A138530 ..11111111111..1011..102..23..21..15..14..13..12..11..10
%e A138530 111111111111..1100..110..30..22..20..15..14..13..12..11..10,
%e A138530 and the triangle of sums of digits starts:
%e A138530 ......................1
%e A138530 .....................2...1
%e A138530 ......... ..........3...2...1
%e A138530 ...................4...1...2...1
%e A138530 ..................5...2...3...2...1
%e A138530 .................6...2...2...3...2...1
%e A138530 ................7...3...3...4...3...2...1
%e A138530 ...............8...1...4...2...4...3...2...1
%e A138530 ..............9...2...1...3...5...4...3...2...1
%e A138530 ............10...2...2...4...2...5...4...3...2...1
%e A138530 ...........11...3...3...5...3...6...5...4...3...2...1
%e A138530 ..........12...2...2...3...4...2...6...5...4...3...2...1.
%t A138530 T[n_, k_] := If[k == 1, n, Total[IntegerDigits[n, k]]];
%t A138530 Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Oct 25 2021 *)
%o A138530 (Haskell)
%o A138530 a138530 n k = a138530_tabl !! (n-1) !! (k-1)
%o A138530 a138530_row n = a138530_tabl !! (n-1)
%o A138530 a138530_tabl = zipWith (map . flip q) [1..] a002260_tabl where
%o A138530    q 1 n = n
%o A138530    q b n = if n < b then n else q b n' + d where (n', d) = divMod n b
%o A138530 -- _Reinhard Zumkeller_, Apr 29 2015
%Y A138530 Cf. A007953. See A240236 for another version.
%Y A138530 Cf. A002260.
%K A138530 nonn,base,tabl
%O A138530 1,2
%A A138530 _Reinhard Zumkeller_, Mar 26 2008